Properties

Label 12.16.a.a
Level 12
Weight 16
Character orbit 12.a
Self dual Yes
Analytic conductor 17.123
Analytic rank 1
Dimension 1
CM No
Inner twists 1

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Newspace parameters

Level: \( N \) = \( 12 = 2^{2} \cdot 3 \)
Weight: \( k \) = \( 16 \)
Character orbit: \([\chi]\) = 12.a (trivial)

Newform invariants

Self dual: Yes
Analytic conductor: \(17.123220612\)
Analytic rank: \(1\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

\(f(q)\) \(=\) \(q \) \(\mathstrut -\mathstrut 2187q^{3} \) \(\mathstrut +\mathstrut 45702q^{5} \) \(\mathstrut +\mathstrut 1217888q^{7} \) \(\mathstrut +\mathstrut 4782969q^{9} \) \(\mathstrut +\mathstrut O(q^{10}) \) \(q \) \(\mathstrut -\mathstrut 2187q^{3} \) \(\mathstrut +\mathstrut 45702q^{5} \) \(\mathstrut +\mathstrut 1217888q^{7} \) \(\mathstrut +\mathstrut 4782969q^{9} \) \(\mathstrut -\mathstrut 26895924q^{11} \) \(\mathstrut -\mathstrut 162581770q^{13} \) \(\mathstrut -\mathstrut 99950274q^{15} \) \(\mathstrut -\mathstrut 743272542q^{17} \) \(\mathstrut -\mathstrut 4003014700q^{19} \) \(\mathstrut -\mathstrut 2663521056q^{21} \) \(\mathstrut -\mathstrut 30097540728q^{23} \) \(\mathstrut -\mathstrut 28428905321q^{25} \) \(\mathstrut -\mathstrut 10460353203q^{27} \) \(\mathstrut +\mathstrut 19021888926q^{29} \) \(\mathstrut -\mathstrut 4621552936q^{31} \) \(\mathstrut +\mathstrut 58821385788q^{33} \) \(\mathstrut +\mathstrut 55659917376q^{35} \) \(\mathstrut +\mathstrut 649297928654q^{37} \) \(\mathstrut +\mathstrut 355566330990q^{39} \) \(\mathstrut +\mathstrut 790230862890q^{41} \) \(\mathstrut +\mathstrut 1388728387532q^{43} \) \(\mathstrut +\mathstrut 218591249238q^{45} \) \(\mathstrut -\mathstrut 3933841180608q^{47} \) \(\mathstrut -\mathstrut 3264310329399q^{49} \) \(\mathstrut +\mathstrut 1625537049354q^{51} \) \(\mathstrut -\mathstrut 13472208095706q^{53} \) \(\mathstrut -\mathstrut 1229197518648q^{55} \) \(\mathstrut +\mathstrut 8754593148900q^{57} \) \(\mathstrut -\mathstrut 24672598493364q^{59} \) \(\mathstrut +\mathstrut 23630686395542q^{61} \) \(\mathstrut +\mathstrut 5825120549472q^{63} \) \(\mathstrut -\mathstrut 7430312052540q^{65} \) \(\mathstrut +\mathstrut 32385083278292q^{67} \) \(\mathstrut +\mathstrut 65823321572136q^{69} \) \(\mathstrut -\mathstrut 74451150070920q^{71} \) \(\mathstrut +\mathstrut 176524276453946q^{73} \) \(\mathstrut +\mathstrut 62174015937027q^{75} \) \(\mathstrut -\mathstrut 32756223088512q^{77} \) \(\mathstrut -\mathstrut 137959485182488q^{79} \) \(\mathstrut +\mathstrut 22876792454961q^{81} \) \(\mathstrut -\mathstrut 458794939458348q^{83} \) \(\mathstrut -\mathstrut 33969041714484q^{85} \) \(\mathstrut -\mathstrut 41600871081162q^{87} \) \(\mathstrut -\mathstrut 32239404369270q^{89} \) \(\mathstrut -\mathstrut 198006386701760q^{91} \) \(\mathstrut +\mathstrut 10107336271032q^{93} \) \(\mathstrut -\mathstrut 182945777819400q^{95} \) \(\mathstrut +\mathstrut 478308097627586q^{97} \) \(\mathstrut -\mathstrut 128642370718356q^{99} \) \(\mathstrut +\mathstrut O(q^{100}) \)

Embeddings

For each embedding \(\iota_m\) of the coefficient field, the values \(\iota_m(a_n)\) are shown below.

For more information on an embedded modular form you can click on its label.

Label \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1
0
0 −2187.00 0 45702.0 0 1.21789e6 0 4.78297e6 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

This newform does not admit any (nontrivial) inner twists.

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(3\) \(1\)

Hecke kernels

This newform can be constructed as the kernel of the linear operator \(T_{5} \) \(\mathstrut -\mathstrut 45702 \) acting on \(S_{16}^{\mathrm{new}}(\Gamma_0(12))\).